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Fermat-Steiner Problem in the Metric Space of Compact Sets endowed with Hausdorff Distanceстатья Электронная публикация
- Авторы: Alexandr Ivanov, Alexandr Tropin, Alexey Tuzhilin
- Журнал: ArXiv e-prints
- Номер: 1601.03852
- Год издания: 2016
- Аннотация: Fermat--Steiner problem consists in finding all points in a metric space $Y$ such that the sum of distances from each of them to the points from some fixed finite subset of $Y$ is minimal. This problem is investigated for the metric space $Y=H(X)$ of compact subsets of a metric space $X$, endowed with the Hausdorff distance. For the case of a proper metric space $X$ a description of all compacts $K\in H(X)$ which the minimum is attained at is obtained. In particular, the Steiner minimal trees for three-element boundaries are described. We also construct an example of a regular triangle in $H(\R^2)$, such that all its shortest trees have no ``natural'' symmetry.
- Добавил в систему: Иванов Александр Олегович